Re: A few questions

Hello Bobbym,Would you mind if I go to bed now?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

Hi Agnishom!

There are other triangles other than right triangles that work. I was just pointing out that there areonly two that are right triangles. The proof of that is probably much easier than the proof for trianglesin general. It will be interesting to see the general proof.

P.S. I tried your link and found the proof. Thanks!

Last edited by noelevans (2012-12-11 09:26:34)

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

Re: A few questions

Hi noelevans!Will you explain the proof?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

Hi! This is Dan's proof of 4 years ago from your link.

Let a,b,c denote the sides of triangle ∆ABC and P and A its perimeter and area respectively.Note that Herons formula states that the area of ∆ABC is: A = √[s(s-a)(s-b)(s-c)]Where s denotes the semi-perimeter, s = (a+b+c)/2.To find all such triangles such that P=A we must have√(a+b+c)(b+c-a)(a+c-b)(a+b-c) = 4(a+b+c).Put x=b+c-a, y=a+c-b and z=a+b-c, where x,y,z are positive integers (true by the triangleinequality). We see that Heron reduces to the Diophantine equation 16(x+y+z)=xyz.

We see from above that x,y,z must all be even integers. So, we may put x=2m, y=2n, z=2k to get,

mnk=4(m+n+k) ⇒ mnk-4m = 4(n+k) ⇒ m = 4(n+k)/(nk-4)

Without loss of generality assume m≥n≥k, then we have 2m≥2n≥n+k ⇒ nk≤12.Since 4<nk≤12 we may test integral values of n and k to find all such triangles.Finally, we see that there is only 5 such triangles:(a,b,c) = (5,12,13), (6,8,10), (6,25,29), (7,15,20), (9,10,17)..................................................................................................................................

I followed pretty easily to the three little lines starting with mnk=4(m+n+k) but had a littledifficulty figuring out why 4<nk≤12. So I went back to the line mnk=4(m+n+k) and got it from there. First rewrite mnk=4(m+n+k) as nk = 4(m+n+k)/m.

Also rewriting mnk=4(m+n+k) as 4 = nk*[m/(m+n+k)] we see that nk is being multipliedby the quantity [m/(m+n+k)] which is less than one. Hence nk>4.

So now we have the 4<nk≤12.

Testing all n≥k being bigger than four but less than or equal to 12 gives us the combinations(n,k) ∈ {(12,1),(6,2),(4,3),(3,2),(4,2),(5,2),(3,3),(6,1),(8,1),(5,1),(7,1), (9,1),(10,1),(11,1)}to test (14 of them).

Testing means substitute the n and k into mnk=4(m+n+k) and solve for m. If the result is apositive integer then double the m, n, and k to get the x, y, and z. Then put these valuesinto the equations x=-a+b+c y= a-b+c z= a+b-c and solve this system for a, b and c (the sides of the triangles).

Example: n=6 and k=1 gives m*6*1=4(m+6+1); that is, 6m=4m+28 so m=14 This case gives an integral value for m so x=2m=28, y=2n=12 and z=2k=2.

Substituting these in to the three equations involving a, b, and c we obtain

28 = -a + b + c 12 = a - b + c 2 = a + b - c

Solving this system by addition/subtraction we obtain a=7, b=15, c=20 which is one of thetriangles that works. (Adding each pair of equations eliminates two variables leaving thethird which is easy to solve.)

A nice solution to the problem, but Dan leaves out a few steps that one must scratch thehead about to fill in the gaps. gotta

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

Re: A few questions

Sorry for my late reply.In the first place, how do you know that if 16(x+y+z) = xyz then x, y and z are even?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

Should I post my next question?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

Hi!

I believe you all are right. The x, y an z do not have to be even, only the product xyz. So onlyone of the three x, y, z has to be even; that is, one or more of x, y, z are even. But I think hisproof essentially holds if his evenness assumption is ignored.

Starting with the 16(x+y+z)=xyz and reasoning in a similar fashion as before for the 4<nk≤12 we can arrive at 16<yz≤48 (assuming z≤y≤x without loss of generality).

Then for y=6 and z=4, 16<yz≤48 and substituting these into 16(x+y+z)=xyz we obtain x=20.Putting these values for x, y and z into the system of equations involving a, b, and cwe obtain a=5, b=12 and c=13 which is one of the triangles that works.

I'll leave it to you to check out the other possibilities for yz to see if the other 4 triangles areobtained.

Edit: P. S. Actually all three of x, y and z must be odd or all three must be even to make a, band c come out to be integers when solving for a, b and c.

If Dan had just left out the sentence: "We see from above that x,y,z must all be even integers."then he could have still made the substitutions x=2m, y=2n and z=2k but readers might wonderwhy he did so. It appears that all that does is change the limits from '16 to 48' to '4 to 12' whichmakes for different pairs to try for solving for x and m, respectively.

Last edited by noelevans (2012-12-15 06:29:47)

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

Re: A few questions

Do you know Dan?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

I think I understand this proof

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

Just googling!Hope it is correctPlease explain me this thing: Why do they assume the sides are in the form of (x+y), (y+z) and (x+z)?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

That means every triangle can be put into that form.How can you say?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

Re: A few questions

Question 9: A triangle has sides of length at most 2, 3 & 4. What is the maximum area the triangle can have?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember